10 SPACES WITH OPERATORS. ORBIT SPACES 47
If E is a Banach space, the group GL(E) of linear homeomorphisms of E
onto E (12.8.1) acts continuously on E by (u, x)i~+u(x) (5.7.4).
Let G be a topological group and E a topological space. The mapping
(s9 x)i-+x of G x E into E is a continuous action of G on E, called the
Let G be a topological group acting continuously on a topological space E.
If p is a continuous homomorphism of a topological group G' into G, then G'
acts continuously on E by (s'y x)\~*p(sf) - x.
(12.10.3) If G is a topological group acting continuously on a topological
space E, then for each s e G the mapping xt-*s - x is a homeomorphism of E
For it is a continuous bijection, and the inverse bijection x^^s"1 • xis
(12.10.4) If G is a topological group acting continuously on a Hausdorff
topological space E, the stabilizer of each point ofE is a closed subgroup ofG.
This follows immediately from (12.3.5).
(12.10.5) Let G be a metrizable group acting continuously on a metrizable
space E. Let A be a compact subset of G, and B a closed (resp. compact)
subset ofE. Then A • B is closed (resp. compact) in E.
The second assertion follows from the fact that A • B is the image of the
compact set A x B (3.20.1 6(v)) by the continuous mapping (s,x)h-*s-x
(3.17.9). As to the first assertion, consider a sequence (sn • xn) of points of
A • B (where sn e A and xn e B) with a limit z e E. By hypothesis, the se-
quence (sn) has a subsequence (snk) converging to some point a e A. Since
xnk = Snkl - (snk • xttk), the sequence (xnj) converges to a""1 • z. But B is closed
in E and therefore a""1 • z e B, hence z = a - (a""1 • z) e A • B. By (3.13.13),
the proof is complete.
Let G be a topological group acting continuously on a topological space
E. Let E/G be the set of orbits and let TC : E -> E/G be the canonical mapping,
so that n(x) is the orbit G - x for each x e E. Let C be the set of subsets U of
E/G such that TiT^U) is open in E. It follows immediately from the formulas
(1.8.5) and (18,6) that D is a topology on E/G. The set E/G, endowed with G, the product group H x K acts continuously